Exponential Form of Complex Numbers

5 Key excerpts on "Exponential Form of Complex Numbers"

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  Mathematical Methods in Engineering and Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  We emphasized the importance of this rule, but did not prove it, because the easiest proof is one quick line of algebra using the exponential form we have now introduced. |z 1 |e i  1 |z 2 |e i  2 = |z 1 ||z 2 |e i ( 1 +  2 ) EXAMPLE Alternative Representations Problem: Express the number z = 3 + 4i in complex exponential form. Solution: The modulus is 5 and the phase is tan −1 (4∕3) ≈ 0.927. We conclude that 3 + 4i = 5e 0.927i . Remember that these are not two different numbers, but two different ways of expressing the same number. The first representation makes it clear that Re (z) = 3 and Im(z) = 4, while the second shows us that |z| = 5 and  z ≈ 0.927. Problem: Express the number 10e (2∕3)i in a + bi form. Solution: Euler’s formula does the conversion for us. 10e (2∕3)i = 10cos 2 3 + 10i sin 2 3 = −5 + 5 √ 3 i 120 Chapter 3 Complex Numbers If you follow the number 3 + ti as t goes from 0 to 1 you trace out a line segment on the complex plane. What does the number 5e it trace out? Consider that question for a moment before proceeding to the next section. The Complex Exponential Function Represents Oscillations In the Motivating Exercise (Section 3.1), a function of the form e ikt came up in solving the equation for a damped oscillator. How does such a function behave? You can trace out the function e it by looking at the “easy” points. Im e i(3π/2) = –i e i(π/2) = i e iπ = –1 e i0 = 1 Re FIGURE 3.2 Adding ∕2 to the argument of a complex exponential rotates the number by 90 ◦ counterclockwise in the complex plane. At t = 0 we have e 0 = 1. (You can see this by plug- ging t = 0 into Euler’s formula, or just remember that anything raised to the zero is 1.) As we discussed above, e (∕2)i = i and e i = −1.Finally, e (3∕2)i = −i .If you’re not seeing a pattern to these numbers, look at the complex plane. We see on Figure 3.2 that the function e it neither grows nor decays, but cycles.

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Complex exponentials 367 definition. If z = x + iy, we define e z to be the complex number given by the equation e z = e x (cos y + i sin y). (9.9) Note that e z = e x when y = 0; hence this exponential agrees with the usual exponential when z is real. Now we shall use this definition to deduce the law of exponents. theorem 9.3. If a and b are complex numbers, we have e a e b = e a+b . (9.10) Proof . Writing a = x + iy and b = u + iv, we have e a = e x (cos y + i sin y), e b = e u (cos v + i sin v), so e a e b = e x e u [cos y cos v − sin y sin v + i(cos y sin v + sin y cos v)]. Now we use the addition formulas for cos (y + v) and sin (y + v) and the law of exponents for real exponentials, and we see that the foregoing equation becomes e a e b = e x+u [cos (y + v] + i sin (y + v)]. (9.11) Since a + b = (x + u) + i(y + v), the right member of (9.11) is e a+b . This proves (9.10). theorem 9.4. Every complex number z ≠ 0 can be expressed in the form z = re i , (9.12) where r = |z| and  = arg (z) + 2n, n being any integer. This representation is called the polar form of z. Proof . If z = x + iy, the polar-coordinate representation (9.5) gives us z = r(cos  + i sin ), where r = |z| and  = arg (z) + 2n, n being any integer. But if we take x = 0 and y =  in (9.9), we obtain the formula e i = cos  + i sin , which proves (9.12). The representation of complex numbers in the polar form (9.12) is especially useful in con- nection with multiplication and division of complex numbers. For example, if z 1 = r 1 e i and z 2 = r 2 e i , we have z 1 z 2 = r 1 e i r 2 e i = r 1 r 2 e i(+) . (9.13) 368 Complex Numbers Therefore the product of the moduli, r 1 r 2 , is the modulus of the product z 1 z 2 , in agreement with Equation (9.6), and the sum of the arguments,  + , is an admissible argument for the product z 1 z 2 . When z = re i , repeated application of (9.13) gives us the formula z n = r n e in = r n (cos n + i sin n), valid for any nonnegative integer n.

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  Linear and Complex Analysis for Applications

  • John P. D'Angelo(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 2 Complex numbers The extent to which complex numbers arise in physics and engineering is almost beyond belief. We start by defining the complex number field ℂ. Since ℂ is a field, in the rest of the book we can and will consider complex vector spaces. In this chapter, however, we focus on the geometric and algebraic properties of the complex numbers themselves. One of the main points is the relationship between the complex exponential function and the trig functions. 1. Basic definitions A complex number is a pair (x, y) of real numbers, or simply the point (x, y) in ℝ 2. By regarding this pair as a single object and introducing appropriate definitions of addition and multiplication, remarkable simplifications arise. As sets, we have ℂ = ℝ 2. We add componentwise as usual, but we introduce a definition of multiplication which has profound consequences. Figures 1 and 2 provide geometric intuition. (x, y) + (a, b) = (x + a, y + b) (1) (x, y) * (a, b) = (x a – y b, x b + y a). (2) The idea behind formula (2) comes from seeking a square root of –1. Write (x, y) = x + iy and (a, b) = a + ib. If we assume the distributive law is valid, then we get (x + i y) (a + i b) = x a + i 2 y b + i (x b + y a). (3) If we further assume in (3) that i 2 = –1, then we get (x + i y) (a + i b) = x a – y b + i (x b + y a) (4) This result agrees with (2). Why do we start with (2) rather than with (4)? The answer is somewhat philosophical. If we start with (2), then we have not presumed the existence of an object whose square is –1. Nonetheless, such an object exists, because (0,1) ∗ (0, 1) = (–1, 0) = –(1, 0). Therefore no unwarranted assumptions have been made. If we started with (4), then one could question whether the number i exists. Henceforth we will use the complex notation, writing (x, y) = x + iy and (a, b) = a + ib. Then (x + i y) + (a + i b) = x + a + i (y + b) (x + i y) (a + i b) = x a – y b + i (x b + y a). Put z = x + iy

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  Story Of Numbers, The

  • Asok Kumar Mallik(Author)
  • 2017(Publication Date)
  • World Scientific Publishing Company

    (Publisher)

  r have the same magnitude. Considering various points lying on a unit circle Wessel even derived the famous De Moivre’s theorem, which states

  4.3.Euler’s Fabulous Formula

  One of the most famous formulas in mathematics, connecting algebra and trigonometry, was produced by Leonhard Euler [25], the wizard of mathematics. Euler considered an imaginary argument iθ for the exponential function. Using equation (3.66a), one writes

  Using the following well-known series expansions for sine and cosine functions one gets the celebrated formula of Euler as For various applications of this formula in different areas of mathematics and other applied sciences see [25].

  Raising both sides of equation (4.16) to the exponent n,

  Thus, we retrieve De Moivre’s theorem expressed by equation (4.13).

  Substituting θ = π in Euler’s formula given by equation (4.16), one obtains

  Expression (4.17) is remarkable as it involves five most important numbers, 0, 1, e, i and π, and two most important mathematical operations with symbols + and =. Famous American physicist Richard Feynman was awe stuck by the beauty of this formula as a 15-year-old student and in his notebook he wrote it as the most beautiful formula in mathematics.

  It should be noted that in view of equation (4.16), a complex number can be expressed in terms of its norm and argument as z = reiθ . It must be pointed out that the argument θ can always be replaced by θ + 2nπ, with n = 0, ±1, ±2, . . .. This fact results in many-valuedness of complex exponentiation and some other functions of complex variables. The value obtained with n

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  ◆◆◆ ◆◆◆ 492 Chapter 21 ◆ Complex Numbers Quotients By Eq. 30: Quotients r e r e r r e j j j 1 2 1 2 ( ) 1 2 1 2 = θ θ θ θ - 234 ◆◆◆ Example 28: (a) e e e 8 4 2 j j j 5 2 3 = (c) e e e 5.82 9.83 0.592 j j j 4 7 3 = - - (b) e e e 63.8 13.7 4.66 j j j 2 5 3 = - Powers and Roots By Eq. 31: Powers and Roots re r e ( ) j n n jn = θ θ 235 ◆◆◆ Example 29: (a) e e (2 ) 16 j j 3 4 12 = (c) e e e e (0. 223 ) (0.223) 0.0497 20.1 j j j j 3 2 6 2 6 6 = = = - - (b) e e ( 3.85 ) 57.1 j j 2 3 6 - = - - - ◆◆◆ ◆◆◆ Exercise 4 ◆ Complex Numbers in Exponential Form Express each complex number in exponential form. 1. j 2 3 + 2. j 1 2 - + 3. j 3(cos 50 sin 50 ) ° + ° 4. 12 14° 5. 2.5 6 π 6. j 7 cos 3 sin 3 π π +       7. 5.4 12 π 8. j 5 4 + Express in rectangular, polar, and trigonometric forms. 9. e 5 j 3 10. e 7 j 5 11. e 2.2 j1.5 12. e 4 j 2 Operations in Exponential Form Multiply. 13. e e 9 2 j j 2 4 ⋅ 14. e e 8 6 j j 3 ⋅ 15. e e 7 3 j j 3 ⋅ 16. e e 6.2 5.8 j j 1.1 2.7 ⋅ 17. e e 1.7 2.1 j j 5 2 ⋅ 18. e e 4 3 j j 7 5 ⋅ Divide. 19. 18e j6 by 6e j3 20. 45e j4 by 9e j2 21. 55e j9 by 5e j6 22. 123e j6 by 105e j2 23. 21e j2 by 7e j 24. 7.7e j4 by 2.3e j2 Evaluate. 25. (3e j5 ) 2 26. (4e j2 ) 3 27. (2e j ) 3 493 Section 21–5 ◆ Vector Operations Using Complex Numbers 21–5 Vector Operations Using Complex Numbers Vectors Represented by Complex Numbers One of the major uses of complex numbers is that they can represent vectors and, as we will soon see, can enable us to manipulate vectors in ways that are easier than we learned when studying oblique triangles. Take the complex number 2 + j3, for example, which is plotted in Fig. 21-7. If we connect that point with a line to the origin, we can think of the complex number 2 + j3 as representing a vector R having a horizontal component of 2 units and a vertical component of 3 units. The complex number used to represent a vector can, of course, be expressed in any of the forms of a complex number.

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